Stochastic invariance for hybrid stochastic differential equation with non-Lipschitz coefficients

Abstract

In this paper, by using of the martingale property and positive maximum principle, we investigate the stochastic invariance for a class of hybrid stochastic differential equations (HSDEs) and provide necessary and sufficient conditions for the invariance of closed sets of $\mathbb{R}^d$ with non-Lipschitz coefficients. Moreover, an example of the most probable phase portrait is given to illustrate the effectiveness of the main results.

Keywords

• linear growth condition
• martingale problem
• hybrid stochastic differential equations
• stochastic invariance